Lagrangian multi-sections and their toric equivariant mirror

Abstract

The SYZ conjecture suggests a folklore that "Lagrangian multi-sections are mirror to holomorphic vector bundles". In this paper, we prove this folklore for Lagrangian multi-sections inside the cotangent bundle of a vector space, which are equivariantly mirror to complete toric varieties by the work of Fang-Liu-Treumann-Zaslow. We also introduce the Lagrangian realization problem, which asks whether one can construct an unobstructed Lagrangian multi-section with asymptotic conditions prescribed by a tropical Lagrangian multi-section. We solve the realization problem for tropical Lagrangian multi-sections over a complete 2-dimensional fan that satisfy the so-called $N$-generic condition with $N\ge 3$. As an application, we show that every rank 2 toric vector bundle on the projective plane is mirror to a Lagrangian multi-section.

31 pages, 7 figures. We add the details of the gluing analysis in Section 5 and an Appendix for Nadler generation result for immersed Lagrangians. The Comments are welcome!

References in corpus (1)

• Tropical Lagrangian multi-sections and toric vector bundles

Cited by in corpus (1)

• Toric vector bundles, non-abelianization, and spectral networks